Approximation in Sobolev spaces and nonlinear potential theory

Structures in circumstellar matter reflect both fast processes and quasi-equilibrium states. A geometrical diversity of emitting circumstellar matter is observed around evolved massive stars, in particular around B[e] supergiants. We recapitulate classical analytical tools of linear and nonlinear potential theory, such as Cole–Hopf transformation and Grad–Shafranov theory, and develop them further to explain the occurrence of the circumstellar matter structures and their dynamics. We use potential theory to formulate the nonlinear hydrodynamical equations and test dilatations of the quasi-equilibrium initial conditions. We find that a wide range of flow patterns can basically be generated and the timescales can switch, based on initial conditions, and lead to eruptive processes, reinforcing that the nonlinear fluid environment includes both quasi-stationary structures and fast processes like finite-time singularities. Some constraints and imposed symmetries can lead to Keplerian orbits, while other constraints can deliver quasi-Keplerian ones. The threshold is given by a characteristic density at the stellar surface.

The classification theory of Riemann surfaces is generalized to Riemanniann-manifolds in the conformally invariant case. This leads to the study of the existence ofA-harmonic functions of typen with various properties and to an extension of the definition of the classical notions with inclusionsOG⊂OHP⊂OHB⊂OHD. In the classical case the properness of the inclusions were proved rather late, in the 50's by Ahlfors and Toki. Our main objective is to show that such inclusions are proper also in the generalized case.

Originally, harmonic morphisms were defined as continuous mappings φ:X → X' between harmonic spaces such that h'∘φ remains harmonic whenever h' is harmonic, see [1], p. 20. In general linear axiomatic potential theory, one has to replace harmonic function

In this paper, we obtain gradient continuity estimates for viscosity solutions of [Formula: see text] in terms of the scaling critical [Formula: see text]-norm of [Formula: see text], where [Formula: see text] is the normalized [Formula: see text]-Laplacian operator. Our main result corresponds to the borderline gradient continuity estimate in terms of the modified Riesz potential [Formula: see text]. Moreover, for [Formula: see text] with [Formula: see text], we also obtain [Formula: see text] estimates. This improves one of the regularity results in [A. Attouchi, M. Parviainen and E. Ruosteenoja, [Formula: see text] regularity for the normalized [Formula: see text]-Poisson problem, J. Math. Pures Appl. (9) 108(4) (2017) 553–591], where a [Formula: see text] estimate was established depending on the [Formula: see text]-norm of [Formula: see text] under the additional restriction that [Formula: see text] and [Formula: see text]. We also mention that differently from the approach in the above paper, which uses methods from divergence form theory and nonlinear potential theory, the method in this paper is more non-variational in nature, and it is based on separation of phases inspired by the ideas in [L. Wang, Compactness methods for certain degenerate elliptic equations, J. Differential Equations 107(2) (1994) 341–350]. Moreover, for [Formula: see text] continuous, our approach also gives a somewhat different proof of the [Formula: see text] regularity result.

Pointwise gradient bounds via Riesz potentials, such as those available for the linear Poisson equation, actually hold for general quasilinear degenerate equations of p-Laplacean type. The regularity theory of such equations completely reduces to that of the classical Poisson equation up to the C1-level.

<div class="htmlview paragraph">This paper describes a software system called FREPS (Forced REsponse Prediction System) that integrates structural dynamic, steady and unsteady aerodynamic analyses to efficiently predict the forced response dynamic stresses in axial flow turbomachinery blades due to aerodynamic and mechanical excitations. A flutter analysis capability is also incorporated into the system. The FREPS system performs aeroelastic analysis by modeling the motion of the blade in terms of its normal modes. The structural dynamic analysis is performed by a finite element code such as MSC/NASTRAN. The steady aerodynamic analysis is based on nonlinear potential theory and the unsteady aerodynamic analysis is based on a linearization of the non-uniform potential mean flow. The program description and presentation of the capabilities are reported herein. The effectiveness of the FREPS package is demonstrated on the High Pressure Oxygen Turbopump turbine of the Space Shuttle Main Engine. Both flutter and forced response analyses are performed and typical results are illustrated.</div>

Nonlinear potential theory in metric spaces

This paper numerically investigates the energy evolution of the focused wave and its correlation with input parameters for the wave generation. The focused wave is numerically generated based on the fully nonlinear potential wave theory which is solved by the Quasi Arbitrary Lagrangian Eulerian-Finite Element Method. The investigation is performed by dividing the generated wave energy into three categories according to their frequency intervals, i.e., the initially assigned frequency interval [f1fN], the frequencies lower (f < f1) and higher (f > fN) than the initial interval. The amount of the generated energy falling into three energy categories and the energy distribution in the initial frequency interval are analysed against amplitude parameters and frequency bands under three amplitude spectra. Four indicators are proposed to indicate the actual energy variations comparing to the initial design. It is found that for all the three frequency intervals the energies increase when the amplitude parameters increases or the given frequency shifts towards the lower frequency domain. The choice of wave amplitude spectrum plays a significant role to maintain the distribution of the energy in the design frequency range and to minimise the energies out of the design frequency range. The results from the second-order wave group theory are used to assist analyses of the mechanism of the energy evolution alongside the focused wave generation. It is found that the lower-frequency energy in the vicinity of the focusing point is dominated by the second-order difference waves, whereas the higher-frequency energy is mainly produced by the wave flap. The waves which are higher than the second-order and the complex interactions not considered by the second-order wave theory are found significantly disturbing the energy distribution in the originally assigned wave-frequency interval.

Hydroelasticity is an important problem in the field of ocean engineering. It can be noted from most of the works published as well as theories proposed earlier that this particular problem was addressed based on the time independent/ frequency domain approach. In this paper, we propose a novel numerical method to address the fluid-structure interaction problem in time domain simulations. The hybrid numerical model proposed earlier for hydro-elasticity (Sriram and Ma, 2012) as well as for breaking waves (Sriram et al 2014) has been extended to study the problem of breaking wave-elastic structure interaction. The method involves strong coupling of Fully Nonlinear Potential Flow Theory (FNPT) and Navier Stokes (NS) equation using a moving overlapping zone in space and Runge kutta 2nd order with a predictor corrector scheme in time. The fluid structure interaction is achieved by a near strongly coupled partitioned procedure. The simulation was performed using Finite Element method (FEM) in the FNPT domain, Particle based method (Improved Meshless Local Petrov Galerkin based on Rankine source, IMPLG_R) in the NS domain and FEM for the structural dynamics part. The advantage of using this approach is due to high computational efficiency. The method has been applied to study the interaction between breaking waves and elastic wall.

A summary of water entry problem of a wedge based on the fully nonlinear velocity potential theory

The explicit solution of some axialsymmetric scalar screen problems in Sobolev spaces is presented. The well-posedness of the boundary integral equations, which are formulated as dual integral equations, is proved. A reduction for the mixed boundary value problem to a system of singular integral equations with a symbol from the Wiener-algebra is given.The approach of dual integral equations has a long history [Mixed Boundary Value Problems in Potential Theory, North-Holland, Amsterdam, 1966] whereas the new developments reviewed in [E. Meister and F. O. Speck, Modern Wiener–Hopf methods in diffraction theory, Proc. Conf. Dundee, 1988, in Ordinary and Partial Differential Equations, B. Sleeman and R. Jarvis, eds., 1989, pp. 130–171] make it possible to handle these equations in Sobolev spaces.

We address the inverse problem for scattering of acoustic waves due to an inhomogeneous medium. We derive and analyze the Hessian in both Hölder and Sobolev spaces. Using an integral equation approach based on Newton potential theory and compact embeddings in Hölder and Sobolev spaces, we show that the Hessian can be decomposed into two components, both of which are shown to be compact operators. Numerical examples are presented to validate our theoretical results. The implication of the compactness of the Hessian is that for small data noise and model error, the discrete Hessian can be approximated by a low-rank matrix. This in turn enables fast solution of an appropriately regularized inverse problem, as well as Gaussian-based quantification of uncertainty in the estimated inhomogeneity.

A solution is derived for a one-dimensional boundary-value problem of nonlinear potential theory with one free end with a supplementary boundary condition. The problem is solved by a variant of the fixed domain method. In each fixed domain, the problem is reduced to the corresponding nonlinear difference problem, which is solved by Newton's method.

A 1D time-varying nonlinear theory based on the Duffing equation is applied to space-charge limited beams and specifically vircators. This theory classifies test-particle trajectories in a modulated nonlinear potential. Two predictions of the theory that can be directly compared to experiment are the final state of electron trajectories and the oscillation frequency of the electrons in the potential well. Experimental measurements of electron flux recorded along the vircator chamber wall correlates well with the numerically integrated final state of electron trajectory in the 1D theory. The oscillation frequency measured in the experiment is shown to be a better match to the oscillation frequency calculated from the nonlinear potential as compared to a parabolic potential (that results from a linear restoring force). In the experiment, random initial conditions arise from beam thermalization and nonuniform electron emission at the surface of the cathode. However, these characteristics alone do not explain the experimentally observed fluctuations in rf power and frequency. The predictions of the time- varying nonlinear potential theory clearly exhibits trends that were observed in the experimental results, in the form of classes of particle trajectories, fluctuations in particle asymptotic states, and particle motion sensitive to the shape of the virtual cathode.© (1995) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

This article concerns with the thermoelastic corner‐edge‐type system with singular potential function on a wedge manifold with corner singularities. First, we introduce weighted p‐Sobolev spaces on manifolds with corner‐edge singularities. Then, we prove the corner‐edge‐type Sobolev inequality, Poincar inequality and Hardy inequality and obtain some results about the compactness of embedding maps on the weighted corner‐edge Sobolev spaces. Finally, as an application of these results, we apply the potential well theory and the Faedo–Galerkin approximations to obtain the global weak solutions for the thermoelastic corner‐edge‐type system 1.1.